The ability to predict average properties of flows through composite materials may have a high impact in many fields of science. Understanding how microscopic behaviors influence macroscopic behaviors on composite materials also will improve understanding of nanostructured materials, turbulence, and phase transitions, including detection of the range of failure of materials. The first step in this direction involves obtaining the effective properties of a composite.
Computing the effective conductivity tensor of a heterogeneous geological formation is important for obtaining accurate numerical simulations on large heterogeneous domains. This is relevant to a number of areas, including conventional and unconventional energy recovery, storage and management of, e.g., oil, water, carbon dioxide, underground wastes and for the study of earthquakes. A fine-scale numerical flow solution involving these applications generally requires orders of 106-109 unknowns, which is infeasible to consider even with modern supercomputers. Therefore, simplified equations given by homogenizing or upscaling these systems are often applied. In such cases, the effective, or upscaled, tensor plays a crucial role, in that it accurately captures the subgrid heterogeneity. Traditionally, the computation involved in obtaining the upscaled or effective tensor for these systems has been accomplished numerically, and various algorithms and methods have been employed for upscaling by homogenization theory. The computational cost involved in obtaining the upscaled tensors, however, may be excessive. Special code design may be required that may also add to computational time and performance.
Because of its importance in reservoir modeling, numerical schemes have been implemented, which explicitly compute the effective coefficient. However, if at each numerical gridcell the heterogeneous function changes (which is the case in practical applications), then the computational cost involved in obtaining the upscaled tensors may overcome the cost of solving the full fine-scale problem.
There exists a need, therefore, for a method of analytically computing the effective tensor in numerical simulators to obtain the effective tensor of a composite (heterogeneous) material, i.e., for an analytical procedure for approximating the effective tensor, K obtained by periodic homogenization, and applicable to general (non-periodic) and locally isotropic media. The motivation for obtaining approximate analytical forms stems from their computational efficiency, which makes such forms attractive for field scale simulations. Such forms are also attractive due their portability for use with existing numerical codes. Moreover, such forms can also be incorporated within numerical multi-scale schemes for speeding up such procedures.
The existence of an effective tensor derived from periodic homogenization and applicable to non-periodic media is based on the mathematical concept of G-convergence. The challenge in obtaining analytically an upscaled tensor applicable to general geometric description of the medium still remains. It can be verified by numerical experiments that, even though the heterogeneous description of the media may be given as locally isotropic, the effective tensor is, in general, anisotropic and full tensor.
Even though exact forms for obtaining an average effective tensor seem to rely on the geometric description of the medium, there are established upper and lower bounds known as the Voigt-Reiss' inequality as expressed by equation (1) below:
                                                        (                                                ∫                  Ω                                ⁢                                                                  ⁢                                                      ⅆ                    x                                                        K                    ⁡                                          (                      x                      )                                                                                  )                                      -              1                                ≤                      K            _                    ≤                                    ∫              Ω                        ⁢                                          K                ⁡                                  (                  x                  )                                            ⁢                                                          ⁢                              ⅆ                x                                                    =                              K            h                    ≤                      K            _                    ≤                      K            a                                              (        1        )            
That is, the effective coefficient is between the harmonic and arithmetic averages, respectively. There are other known bounds, such as, for example, Hashin & Shtrikman, which is applicable to two-phase media only, and the Cardwell and Parson bounds. This latter formulation, however, has been demonstrated to be stricter than the equations above and also with a broader applicability.